An analysis of the structure of trees and graphs

Abstract

This thesis is concerned with the structure of trees and linear graphs. In particular an attempt is made to relate structure of these objects to the known methods for counting them. Although the work described here is essentially not computer oriented, the generation and decomposition of graphs and trees by computer is in the background, and so a short section on the computer representation of the various objects is included. Trees are analysed bearing in mind the counting methods due first to Cayley, and a later method using Polya's classical theorem of enumerative combinatorial analysis. Various methods of representation and generation of trees are presented and compared. This thesis then goes on to the substantially more difficult problem of analysing graphs using similar techniques, and attempts are made to relate the structure of graphs to the known techniques for enumerating graphs. This involves a more detailed study of Polya's theorem and an investigation into the underlying concepts such as permutation groups as they are applicable to the case under scrutiny. Representations are developed to aid these investigations. In the following section of the work, methods are investigated for the decomposition of a linear graph, and a number of different decompositions or factorisations are looked at. One such factorisation considered in some detail is the problem of extracting a spanning tree from a grapho and the ways in which the remaining graph or co-tree graph may be manipulated. The complete decomposition of a graph into trees may be achieved using these methods, and the concept of the structure tree of the decomposition is introduced and its properties explored. The techniques described have all been implemented, and a discussion of the problems of the implementation together with some estimates of timing reguirements is also included.